When do jumps matter for portfolio optimization?

Abstract

We consider the continuous-time portfolio optimization problem of an investor with constant relative risk aversion who maximizes expected utility of terminal wealth. The risky asset follows a jump-diffusion model with a diffusion state variable. We propose an approximation method that replaces the jumps by a diffusion and solve the resulting problem analytically. Furthermore, we provide explicit bounds on the true optimal strategy and the relative wealth equivalent loss that do not rely on quantities known only in the true model. We apply our method to a calibrated affine model. Our findings are threefold: Jumps matter more, i.e. our approximation is less accurate, if (i) the expected jump size or (ii) the jump intensity is large. Fixing the average impact of jumps, we find that (iii) rare, but severe jumps matter more than frequent, but small jumps. JEL-Classification AbstractWe consider the continuous-time portfolio optimization problem of an investor with constant relative risk aversion who maximizes expected utility of terminal wealth. The risky asset follows a jump-diffusion model with a diffusion state variable. We propose an approximation method that replaces the jumps by a diffusion and solve the resulting problem analytically. Furthermore, we provide explicit bounds on the true optimal strategy and the relative wealth equivalent loss that do not rely on quantities known only in the true model. We apply our method to a calibrated affine model. Our findings are threefold: Jumps matter more, i.e. our approximation is less accurate, if (i) the expected jump size or (ii) the jump intensity is large. Fixing the average impact of jumps, we find that (iii) rare, but severe jumps matter more than frequent, but small jumps. JEL-Classification: G11; C63

Co-jumps and recursive preferences in portfolio choices

“…By indicating with V and π (resp., Ṽ and π) the value function and the optimal weight for the true model (resp., the approximate model), we can evaluate the Relative WEL (RWEL), i.e. the percentage of initial wealth that can be sacrificed by an investor using the optimal strategy, in order to have the same indirect utility using the approximate one, see Ascheberg et al (2016) for further details. The RWEL is obtained by equating…”

“…It is worth noting that co-jumps' presence produces a strongly non-linear Hamilton-Jacobi-Bellman equation, the latter being unable to be solved in an analytical form. Hence, we impose a linearisation condition for the jump component, as in Ascheberg et al (2016) and Oliva and Renò (2018), and a (log-)linear expansion for the normalised aggregator of current consumption and continuation utility, as in Chacko and Viceira (2005), ensuring an approximate solution. We further analyse the economic implications of our theoretical results on real data, taking into account the investors' characteristics.…”

Consuming durable goods when stock markets jump: A strategic asset allocation approach

“…The impact of jumps on stock markets investment was also studied by Liu et al (2003), who conclude that jumps can lead to a substantial decrease in stock market investment, particularly for individuals with low risk-aversion. The equivalent wealth losses from ignoring jumps were estimated in Ascheberg et al (2013) and in Das and Uppal (2004). These authors conclude that these losses are generally small, but they can be substantial for individuals with a relatively low risk-aversion.…”